\(\mathbb {J}\)  is the Jacobian of the transformation. We know from [1]} that the noncommutative anomaly in two-dimensions is \(\epsilon ^{\mu \nu }\hat{F}_{\mu \nu }(x)\)  up to a constant , where \(\hat{F}_{\mu \nu }\)  is the noncommutative two-forms field strengthThe noncommutative field strength is the ordinary two-forms field strength plus some terms in order of \(\theta \)  which we call it the antisymmetric constant tensor[2]}, [3]}, [4]}.The first order corrections to the one-form noncommutative vector field and two-forms noncommutative field strength are respectively
\(A^{1}_{\gamma }=-\frac{1}{4}\theta ^{\kappa \lambda }\big \lbrace A_{\kappa },\partial _{\lambda }A_{\gamma }+F_{\lambda \gamma }\big \rbrace \) 
\(F^{1}_{\gamma \rho }=-\frac{1}{4}\theta ^{\kappa \lambda }\big ( \big \lbrace A_{\kappa },\partial _{\lambda }F_{\gamma \rho }+D_{\lambda }F_{\gamma \rho }\big \rbrace -2\big \lbrace F_{\gamma \kappa },F_{\rho \lambda }\big \rbrace  \big )\) 
; thus the divergence of Chiral current is
\( \langle D_{\mu }\; \hat{j^{a\; \mu }_{3}}(x) \rangle = \partial _{\mu }\langle \hat{j^{a\;\mu }_{3}}(x)\rangle + f^{abc}\;\hat{A^{b}_{\mu }}(x)\star \langle \hat{j^{c\mu }_{3}}(x)\rangle =- \frac{i}{4\pi }\epsilon ^{\mu \nu }\;\hat{F^{a}_{\mu \nu }}(x)\) 
